The equation | z - i | = | z - 1 | , i = - 1 , represents:

the line through the origin with slope 1

the line through the origin with slope - 1

If z is a complex number such that z ≥ 2 , then the minimum value of z + 1 2 :

Is strictly greater than 3 2 but less than 5 2

Let a , b ∈ R and a 2 + b 2 ≠ 0 . Suppose S = z ∈ C : z = 1 a + i b t , t ∈ R , t ≠ 0 , where i = - 1 . If z = x + i y and z ∈ S , then x , y lies on

The y - axis for a = 0 , b ≠ 0

The circle with radius - 1 2 a and centre - 1 2 a , 0 for a < 0 , b ≠ 0

MEDIUM

Earn 100

If the imaginary part of $\frac{ω}{ω^{'}}$ where $ω$ and $ω^{'}$ are complex numbers be zero, show that $ω$ and $ω^{'}$ lie on a straight line passing through the origin.

Important Questions on Complex Numbers

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

The value of

λ

for which the loci

\arg z = \frac{π}{6}

and

| z - 2 \sqrt{3} i | = λ

on the argand plane touch each other is

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

z \in C

be such that

|z| < 1 .

ω = \frac{5 + 3 z}{5 (1 - z)},

then:

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

S

be the set of all complex numbers

z

satisfying

|z - 2 + i| \geq \sqrt{5} .

If the complex number

z_{0}

is such that

\frac{1}{|z_{0} - 1|}

is the maximum of the set

\{\frac{1}{|z - 1|} : z \in S\},

then the principal argument of

\frac{4 - z_{0} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} + 2 i}

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let the point

P

represent

z = x + i y, x, y \in R in

the Argand plane. Let the curves

C_{1}

and

C_{2}

be the loci of

P

satisfying the conditions

(i)

\frac{2 z + i}{z - 2}

is purely imaginary and

(ii)

Arg (\frac{z + i}{z + 1}) = \frac{π}{2}

, respectively. Then the point of intersection of the curves

C_{1}

and

C_{2},

other than the origin, is

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let complex numbers

α

and

\frac{1}{\bar{α}}

lie on circles

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}

, and

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}

, respectively. If

z_{0} = x_{0} + i y_{0}

satisfies the equation

2 {|z_{0}|}^{2} = r^{2} + 2

, then

|α| =

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Z

is a complex number such that

|Z| \leq 2

and

- \frac{π}{3} \leq amp Z \leq \frac{π}{3}

. The area of the region formed by locus of

Z

is (in sq. units)

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

The equation

| z - i | = | z - 1 |, i = \sqrt{- 1},

represents:

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let the lines

(2 - i) z = (2 + i) \bar{z}

and

(2 + i) z + (i - 2) \bar{z} - 4 i = 0,

(here

i^{2} = - 1

) be normal to a circle

C

. If the line

i z + \bar{z} + 1 + i = 0

is tangent to this circle

C

, then its radius is :

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

z \in C,

the set of complex numbers. Then the equation,

2 |z + 3 i| - |z - i| = 0

represents:

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

The equation

I m (\frac{i z - 2}{z - i}) + 1 = 0, z \in C, z \neq i

represents a part of a circle having radius equal to :

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

A (3 - i), B (2 + i)

be two points in the Argand plane. If the point

P

represents the complex number

z = x + i y,

which satisfies

|z - 3 + i| = |z - 2 - i|,

then the locus of the point

P

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

z_{1}

and

z_{2}

be two complex numbers satisfying

| z_{1} | = 9

and

| z_{2} - 3 - 4 i | = 4

. Then the minimum value of

|z_{1} - z_{2}|

is :

EASY

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

The equation

z \bar{z} + (2 - 3 i) z + (2 + 3 i) \bar{z} + 4 = 0

represents a circle of radius

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Suppose

z \in C

has argument

θ

such that

0 < θ < \frac{π}{2}

and satisfy the equation

| z - 3 i | = 3

, then what is the value of

\cot θ - \frac{6}{z} ?

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

z = x + i y

and

w = u + i v

be complex numbers on the unit circle such that

z^{2} + w^{2} = 1

. Then the number of ordered pairs

(z, w)

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

If the equation

a | z |^{2} + \bar{\bar{α} z + α \bar{z}} + d = 0

represents a circle where

a, d

are real constants then which of the following condition is correct?

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

z

be those complex numbers which satisfy

|z + 5| \leq 4

and

z (1 + i) + \bar{z} (1 - i) ⩾ - 10, i = \sqrt{- 1}

. If the maximum value of

| z + 1 |^{2}

α + β \sqrt{2}

, then the value of

(α + β)

MEDIUM

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

z

is a complex number such that

|z| \geq 2,

then the minimum value of

|z + \frac{1}{2}|

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

The largest value of

r

, for which the region represented by the set

\{ω \in C| |ω - 4 - i| \leq r\}

is contained in the region represented by the set

\{z \in C| |z - 1| \leq |z + i|\}

, is equal to :

HARD

Mathematics>Algebra>Complex Numbers>Geometry of Complex Numbers

Let

a, b \in R

and

a^{2} + b^{2} \neq 0

. Suppose

S = \{z \in C : z = \frac{1}{a + i b t}, t \in R, t \neq 0\},

where

i = \sqrt{- 1} .

z = x + i y

and

z \in S,

then

(x, y)

lies on

If the imaginary part of ωω' where ω and ω' are complex numbers be zero, show that ω and ω' lie on a straight line passing through the origin.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

If the imaginary part of $\frac{ω}{ω^{'}}$ where $ω$ and $ω^{'}$ are complex numbers be zero, show that $ω$ and $ω^{'}$ lie on a straight line passing through the origin.